DLMF:14.25.E1 (Q4958)

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DLMF:14.25.E1
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    Statements

    Defining formula
    P ν - μ ⁡ ( z ) = ( z 2 - 1 ) μ / 2 2 ν ⁢ Γ ⁡ ( μ - ν ) ⁢ Γ ⁡ ( ν + 1 ) ⁢ ∫ 0 ∞ ( sinh ⁡ t ) 2 ⁢ ν + 1 ( z + cosh ⁡ t ) ν + μ + 1 ⁢ d t , Legendre-P-first-kind 𝜇 𝜈 𝑧 superscript superscript 𝑧 2 1 𝜇 2 superscript 2 𝜈 Euler-Gamma 𝜇 𝜈 Euler-Gamma 𝜈 1 superscript subscript 0 superscript 𝑡 2 𝜈 1 superscript 𝑧 𝑡 𝜈 𝜇 1 𝑡 {\displaystyle{\displaystyle P^{-\mu}_{\nu}\left(z\right)=\frac{\left(z^{2}-1% \right)^{\mu/2}}{2^{\nu}\Gamma\left(\mu-\nu\right)\Gamma\left(\nu+1\right)}% \int_{0}^{\infty}\frac{(\sinh t)^{2\nu+1}}{(z+\cosh t)^{\nu+\mu+1}}\mathrm{d}t% ,}}
    0 references
    DLMF id
    DLMF:14.25.E1
    0 references
    constraint
    ℜ ⁡ μ > ℜ ⁡ ν > - 1 𝜇 𝜈 1 {\displaystyle{\displaystyle\Re\mu>\Re\nu>-1}}
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    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2adec
    0 references
    Q11481
    Defining formula
    P ν μ ⁡ ( z ) Legendre-P-first-kind 𝜇 𝜈 𝑧 {\displaystyle{\displaystyle P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C14.S21.SS1.p1.m1adec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
    0 references
    hyperbolic cosine function
    Defining formula
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2adec
    0 references
    hyperbolic sine function
    Defining formula
    sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}
    xml-id
    C4.S28.E1.m2adec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1adec
    0 references
    Q11638
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C14.S1.XMD4.m1dec
    0 references
    Q11591
    Defining formula
    μ 𝜇 {\displaystyle{\displaystyle\mu}}
    xml-id
    C14.S1.XMD7.m1dec
    0 references
    Q11590
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C14.S1.XMD8.m1dec
    0 references
     
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